Location of fault on series-compensated power transmission lines

ABSTRACT

The present invention relates to a method for locating a fault on a power transmission line which is series-compensated with capacitors. The site of the fault is determined with the aid of measured currents and voltages at one of the two ends of the power transmission line before and after the occurrence of the fault and a calculating algorithm based on impedance models of the power transmission line before and after the occurrence of the fault.

TECHNICAL FIELD

The present invention forms an integral part of a protection system for power transmission lines which are series-compensated with capacitors. Specifically, the invention relates to a method and a device for locating a fault which has occurred on the power transmission line. The distance from a station, located at one end of the power transmission line, to the fault is determined with the aid of calculating algorithms based on impedance models of the power transmission line and voltages and currents measured at the station before and after the occurrence of the fault.

BACKGROUND ART, PROBLEMS

A method for fault location on a three-phase power transmission line is known from U.S. Pat. No. 4,559,491. The method for fault location on a three-phase power transmission line, described in U.S. Pat. No. 4,559,491, is based on an impedance model of the power transmission line. A condition for the method according to U.S. Pat. No. 4,559,491 is that the power transmission line has a linear impedance also when a fault has occurred. Series capacitors for phase compensation are normally provided with overvoltage protection devices in the form of some protection component with a non-linear characteristic in the area of interference and the method according to U.S. Pat. No. 4,559,491 is therefore not applicable to a three-phase power transmission line with series capacitors for phase compensation.

From U.S. Pat. No. 4,719,580 it is known that fault location on a three-phase power transmission line may be made according to method based on a waveguide model of the power transmission line. Fault location on a power transmission line with series capacitors for phase compensation based on a waveguide model of the power transmission line may, for certain types of faults, give a misleading result and this fault-location technique has therefore not been accepted by the users.

A method for fault location based on an impedance model of the power transmission line should take into consideration the fact that the occurrence of a fault on the monitored power transmission line entails the feeding of electric power from both ends of the power transmission line to the site of the fault in question since otherwise the fault location cannot be performed with sufficient accuracy. To obtain a result with satisfactory accuracy in the fault-position determination, fault location based on an impedance model of the power transmission line assumes that the power inputs from the two ends of the power transmission line are determined by measuring or calculating the two power inputs. From U.S. Pat. No. 5,455,776 it is known that fault location on a three-phase transmission line may be made with a method which assumes measurement of voltage and current at the two ends of the monitored transmission line. The method could be adapted to series-compensated transmission lines but has the disadvantage of being dependent on measured values from the two ends of the transmission line, which complicates the fault location and makes it more expensive compared with methods operating with measured values from one end of the transmission line only, normally the end of the transmission line where the fault location is arranged.

A method for fault location based on measurement at one end of the transmission line only is described in IEEE Transaction on Power Apparatus and Systems, Vol. PAS-104, No. 2, February 1985, pp. 424-436. This method assumes that the monitored part of the transmission line is described by a linear model and the method is thus not useful for series-compensated transmission lines with non-linear protection devices for protecting the capacitors against overvoltage.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an impedance model of a faultless power transmission line, series-compensated with capacitors with overvoltage protection device, between two stations A and B.

FIG. 2 shows an impedance model of the same power transmission line when a fault has occurred between the series capacitors and station B.

FIG. 3 shows an impedance model of the same power transmission line when a fault has occurred between station A and the series capacitors.

FIG. 4 shows possible fault configurations on the power transmission line.

FIGS. 5 and 6 show how the parallel connection of the series capacitors and the overvoltage protection device may be replaced by a model-equivalent impedance consisting of a series connection of a resistance and a reactance.

FIG. 7 shows how the parameters of the model equivalent as a function of a current, passing through the impedance, may be determined.

FIG. 8 shows how a device for carrying out the method may be designed.

SUMMARY OF THE INVENTION, ADVANTAGES

A single-line diagram for a faultless power transmission line, series-compensated with capacitors, between two stations A and B is shown in FIG. 1. In this context, the terms “backward” and “forward” are often used when referring to the location of a power network in relation to the power transmission line. A backward power network according to FIG. 1 is represented by an emf E_(A) and an impedance Z_(A). In a corresponding way, a forward network is represented by an emf E_(B) and an impedance Z_(B). Further, FIG. 1 shows

Z_(L) the impedance of the power transmission line

d the distance in per unit from station A to the series capacitors

dZ_(L) the impedance of the power transmission line between station A and the series-compensated capacitors.

The capacitors in series-compensated power networks are normally provided with a parallel-connected overvoltage protection device. The parallel connection will be referred to below as “SC&OVP” and is represented by a non-linear impedance because of the non-linear current/voltage characteristic of the overvoltage protection device.

Z_(SC) non-linear impedance consisting of the series capacitors and their overvoltage protection devices

(1-d)dZ_(L) the impedance of the power transmission line between SC&OVP and station B.

The faultless power transmission line, that is, prior to the occurrence of a fault, may thus, according to FIG. 1, be described, in matrix form, by the equation

E _(A) −E _(B) =[Z _(A) +Z _(B) +Z _(L) +Z _(SC) _(—) _(pre) ]I _(A) _(—) _(pre)  (1)

The index “pre” implies that the value of the quantity in question is the value prior to the occurrence of a fault. In the following description, equations with quantities with the index “post” will also be described, which, in a corresponding manner, implies that the value of the quantity in question is the value after the occurrence of a fault. A condition for the distance-to-fault calculation is thus that a continuous measurement and storage of a number of consecutively measured voltage and current values is performed.

As regards the post-fault situation, it must be distinguished whether the fault is located between station A and SC&OVP or if the fault is located between SC&OVP and station B.

FIG. 2 shows an impedance model of a series-compensated power transmission line when a fault has occurred between SC&OVP and station B. A fault which has occurred within this area will be referred to below as a fault 1. The impedance model according to FIG. 2 may, in matrix form, be described with the equation

E _(A) −E _(B) =[Z _(A) +x ₁ Z _(L) +Z _(SC) _(—) _(post) ]I _(A) _(—) _(post)−[(1−x ₁)Z _(L) +Z _(B) ]I _(B) _(—) _(post)  (2)

where x₁ designates the distance from the end point A of the monitored power transmission line to the fault.

FIG. 3 shows an impedance model and a series-compensated power transmission line when a fault has occurred between station A and SC&OVP. A fault which has occurred within this area will be referred to below as a fault 2. The impedance model according to FIG. 3 may, in matrix form, be described with the equation

E _(A) −E _(B) =[Z _(A) +x ₂ Z _(L) ]I _(A) _(—) _(post)−[(1−x ₂)Z _(L) +Z _(B) +Z _(SC) _(—) _(post) ]I _(B) _(—) _(post)  (3)

In FIGS. 2 and 3, the fault is represented by a fault resistance R_(f) to ground or zero. A fault on a three-phase line R, S and T may occur in a plurality of different ways. FIG. 4 shows how a fault with a fault resistance R_(R), R_(S) and R_(T) may occur between any of the phases and ground and as R_(RS), R_(ST) etc. between different phases. A combination of the individual cases may also occur. The voltage at the fault site is defined as V_(fR), V_(fS) and V_(fT) and the corresponding currents as I_(fR), I_(fS) and I_(fT.)

To be able to distinguish the different types of fault that may occur, the distance-to-fault calculation makes use of a fault-type matrix which has the general form $K_{f} = \left\lbrack \begin{matrix} k_{RR} & k_{SR} & k_{TR} \\ k_{RS} & k_{SS} & k_{TS} \\ k_{RT} & k_{ST} & k_{TT} \end{matrix}\quad \right\rbrack$

The elements in K_(f) are determined in dependence on the type of fault as follows:

non-diagonal elements are given the value 0 if the phase in question is not concerned by the relevant fault and the value −1 if the phase is concerned by the relevant fault

The diagonal elements are given the value 1 if the phase in question has a fault to ground at the fault in question and to this is added the sum of the absolute values of the non-diagonal elements in the relevant line.

A few examples of a filled-in matrix K_(f) for some typical types of fault are shown below $K_{f} = {\left\lbrack \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}\quad \right\rbrack \quad {for}\quad {fault}\quad R\text{-}{ground}}$ ${K_{f} = {\left\lbrack \quad \begin{matrix} 2 & {- 1} & {- 1} \\ {- 1} & 2 & {- 1} \\ {- 1} & {- 1} & 2 \end{matrix}\quad \right\rbrack \quad {for}\quad {fault}\quad R\text{-}S\text{-}T}}\quad$ $K_{f} = {\left\lbrack \quad \begin{matrix} 3 & {- 1} & {- 1} \\ {- 1} & 3 & {- 1} \\ {- 1} & {- 1} & 3 \end{matrix}\quad \right\rbrack \quad {for}\quad {fault}\quad R\text{-}S\text{-}T\text{-}{ground}}$ $K_{f} = {\left\lbrack \quad \begin{matrix} 1 & {- 1} & 0 \\ {- 1} & 1 & 0 \\ 0 & 0 & 0 \end{matrix}\quad \right\rbrack \quad {for}\quad {fault}\quad R\text{-}S}$ ${K_{f} = {\left\lbrack \quad \begin{matrix} 2 & {- 1} & 0 \\ {- 1} & 2 & 0 \\ 0 & 0 & 0 \end{matrix}\quad \right\rbrack \quad {for}\quad {fault}\quad R\text{-}S\text{-}{ground}}}\quad$

The vectorial relationship between the fault current I_(f), i.e. I_(fR), I_(fS) and I_(fT), the fault voltage V_(f), i.e. V_(fR), V_(fS) and V_(fT), the equivalent fault resistance R_(f) and the fault matrix is clear from the equation

I _(f) =K _(f) V _(f) /R _(f)  (4)

In connection with the calculation of the distance to the fault (hereinafter referred to as the distance-to-fault calculation) which is described below, knowledge of Z_(SC) is needed, that is, the value of the non-linear impedance as a function of the actual current flowing through the impedance. One way of determining Z_(SC) is to transform the parallel connection according to FIG. 5 into a model-equivalent impedance according to FIG. 6 in the form of a series connection of a resistance R_(SC) and a reactance X_(SC). By, for example, a transient test, R_(SC) and X_(SC) may be determined individually as functions of a traversing current I_(SC), according to FIG. 7, or in the form of a corresponding analytic function. This implies that, when the current through Z_(SC) is known, Z_(SC) may be determined as

Z _(SC) =R _(SC) +jX _(SC)  (5)

In the case of a fault 1 according to FIG. 2 and equations (2), both R_(SC) and X_(SC), i.e. Z_(SC), can be determined with the current I_(A) measured in station A.

In case of a fault 2 according to FIG. 3 and equation (3), however, it is the unknown current I_(B) which traverses Z_(SC). It is, therefore, an integral part of the invention to obtain a measure of the current I_(B) by means of an iterative process. This current is then calculated by means of equation (6) with an assumption of Z_(SC) and the distance x₂.

I _(B)=((1−x ₂)Z _(L) +Z _(B) +Z _(SC))⁻¹.((Z _(A) +x ₂ Z _(L))I _(A)−(E _(A) −E _(B)))  (6)

The value of I_(B) thus obtained determines, in its turn, a preliminary value of Z_(SC) in the same way as above and will be used in the calculating algorithm which will be described below for the distance-to-fault calculation.

The calculating algorithms according to the invention which calculate the distances from station A to the fault are based on the impedance models related above and corresponding equations in matrix form. Since the impedance models are different depending on whether the fault is located between SC&OVP and station B or between station A and SC&OVP, also the calculating algorithms will be different.

The calculating algorithm for calculating a fault 1, that is, calculating the distance from station A to a fault which is located between SC&OVP and station B, consists of the equations

E _(A) −E _(B) =[Z _(A) +Z _(B) +Z _(L) +Z _(SC) _(—) _(pre) ]I _(A) _(—) _(pre)  (1)

E _(A) −E _(B) =[Z _(A) +x ₁ Z _(L) +Z _(SC) _(—) _(post) ]I _(A) _(—) _(post)−[(1−x ₁)Z _(L) +Z _(B) ]I _(B) _(—) _(post)  (2)

V _(A) _(—) _(post) −V _(f)=(x ₁ Z _(L) +Z _(SC) _(—) _(post))I _(A) _(—) _(post)  (7)

I _(f) =I _(A) _(—) _(post) +I _(B) _(—) _(post)  (8)

where x₁ designates the unknown distance from the end point A of the monitored transmission line to the site of the fault, and V_(f) is obtained by means of equation (4).

If (1) is combined with (2, 4, 7, 8), a first distance-to-fault equation is obtained, after simplification, according to the following scalar quadratic equation

a ₁ x ₁ ² −b ₁ x ₁ +c ₁ −R _(f1)=0  (9)

where a ₁, b ₁ and c ₁ designate complex coefficients. By means of a first auxiliary vector

D ₁=(Z _(A) +Z _(B) +Z _(L))(I _(A) _(—) _(post) −I _(A) _(—) _(pre))+Z _(SC) _(—) _(post) I _(A) _(—) _(post) −Z _(SC) _(—) _(pre) I _(A) _(—) _(pre)  (10)

and

D₁ ^(T) which designates the transpose matrix of D₁

The coefficients a₁, b₁ and c₁ may be calculated as follows $\begin{matrix} {{\underset{\_}{a}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}Z_{L}K_{f}Z_{L}I_{A\_ post}}} & (11) \\ {{\underset{\_}{b}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}\quad \left( {{Z_{L}{K_{f}\left\lbrack {V_{A\_ post} + {\left( {Z_{L} - Z_{SC\_ post}} \right)I_{A\_ post}}} \right\rbrack}} + {Z_{B}K_{f}Z_{L}I_{A\_ post}}} \right)}} & (12) \\ {{\underset{\_}{c}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}\left( {Z_{L} + Z_{B}} \right)\left( {V_{A\_ post} - {Z_{SC\_ post}I_{A\_ post}}} \right)K_{f}}} & (13) \end{matrix}$

The first distance-to-fault equation is solved by means of an iterative process which is to be described in greater detail below. of the two solutions of the quadratic equation, only one gives a practical solution, namely a distance to fault equal to $\begin{matrix} {x_{11} = \frac{{{Im}\left( {\underset{\_}{b}}_{1} \right)} - \sqrt{{{{Im}\left( {\underset{\_}{b}}_{1} \right)}{{Im}\left( {\underset{\_}{b}}_{1} \right)}} - {4{{Im}\left( {\underset{\_}{a}}_{1} \right)}{{Im}\left( {\underset{\_}{c}}_{1} \right)}}}}{2{{Im}\left( {\underset{\_}{a}}_{1} \right)}}} & (14) \end{matrix}$

where Im(a₁), Im(b₁) och Im(c₁) are equal to the imaginary part of the coefficients.

It has been found out that the second root of the distance-to-fault equation (9) gives unreasonable values of the distance to the fault and that the equation (14) therefore represents a sought solution to the first distance-to-fault equation (9).

By inserting the obtained value x₁₁ into the first distance-to-fault equation (9), a corresponding fault resistance R_(f1) is obtained by the relationship

R _(f1) =Re( a ₁)x ₁₁ ² −Re( b ₁)x ₁₁ +Re( c ₁)  (15)

where Re(a₁), Re(b₁) and Re(c₁) are equal to the real part of the coefficients.

Determination of distance to fault and fault resistance is made by an iterative process which will be described below.

The calculating algorithm for the distance-to-fault calculation in case of fault 2 consists of the following equation system:

E _(A) −E _(B) =[Z _(A) +Z _(B) +Z _(L) +Z _(SC) _(—) _(pre) ]I _(A) _(—) pre  (1)

 E _(A) −E _(B) =[Z _(A) +x ₂ Z _(L) ]I _(A) _(—) _(post)−[(1−x ₂)Z _(L) +Z _(B) +Z _(SC) _(—) _(post) ]I _(B) _(—) _(post)  (3)

V _(A) _(—) _(post) −V ₁ =x ₂ Z _(L) I _(A) _(—) _(post)  (16)

I _(f) =I _(A) _(—) _(post) +I _(B) _(—) post  (17)

where x₂ designates the unknown distance from the end point A of the monitored transmission line to the site of the fault, and V_(f) is obtained by means of equation (4). If (1) is combined with (3, 5, 16, 17), a second distance-to-fault equation is obtained, after simplification, according to the following scalar quadratic equation with x₂ as variable

a ₂ x ₂ ² −b ₂ x ₂ +c ₂ −R _(f2)=0  (18)

where a ₂, b ₂ and c ₂ designate complex coefficients. By means of a second auxiliary vector

D ₂=(Z _(A) +Z _(B) +Z _(L))(I _(A) _(—) _(post) −I _(A) _(—) _(pre))+Z _(SC) _(—) _(post) I _(A) _(—) _(post) −Z _(SC) _(—) _(pre) I _(A) _(—) _(pre)  (19)

and

D₂ ^(T) which designates the transpose matrix of D₂

The coefficients a₂, b₂ and c₂ may be calculated as follows: $\begin{matrix} {{\underset{\_}{a}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}Z_{L}K_{f}Z_{L}I_{A\_ post}}} & (20) \\ {{\underset{\_}{b}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}\quad \left( {{Z_{L}K_{f}V_{A\_ post}} + {\left\lbrack {Z_{L} + Z_{SC\_ post} + Z_{B}} \right\rbrack K_{f}Z_{L}I_{A\_ post}}} \right)}} & (21) \\ {{\underset{\_}{c}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}\left( {Z_{L} + Z_{B} + Z_{SC\_ post}} \right)K_{f}V_{A\_ post}}} & (22) \end{matrix}$

The solution to the second distance-to-fault equation (16) gives a sought distance to the site of the fault as $\begin{matrix} {x_{21} = \frac{{{Im}\left( {\underset{\_}{b}}_{2} \right)} - \sqrt{{{{Im}\left( {\underset{\_}{b}}_{2} \right)}{{Im}\left( {\underset{\_}{b}}_{2} \right)}} - {4{{Im}\left( {\underset{\_}{a}}_{2} \right)}{{Im}\left( {\underset{\_}{c}}_{2} \right)}}}}{2{{Im}\left( {\underset{\_}{a}}_{2} \right)}}} & (23) \end{matrix}$

It has been found out that the second root of the distance-to-fault equation (18) gives unreasonable values of the distance to the fault and that the equation (23) therefore represents a sought solution to the first distance-to-fault equation (18).

By inserting the obtained value x₂₁ into the second distance-to-fault equation (18), the corresponding fault resistance R_(f2) is obtained by the relationship

R _(f2) =Re( a ₂)x ₂₁ ² −Re( b ₂)x ₂₁ +Re( c ₂)  (24)

Determination of distance to fault and fault resistance is made by an iterative process which will be described below.

Before the iterative process for calculating the distance to fault can be started, a number of conditions must be fulfilled:

Independently of whether the fault is a category 1 fault or a category 2 fault, information about the value of fixed parameters, included in the equation systems, must be available. This relates to values of the backward and forward power networks, the length of the power transmission line, the impedance, the distance from station A to the series capacitors, and so on.

An assumption must be made on an initial value of distance to fault and fault resistance. This assumption may be made with arbitrary values within permissible limits.

When a fault has occurred, the type of fault and hence also the K_(f) matrix may be determined.

To be able to determine which of the calculating algorithms is to be used, knowledge must be gained as to on which side of SC&OVP the fault has occurred, that is, whether it is a fault 1 or a fault 2.

There must be a continuous access, both before and after a fault has occurred, of filtered values of current and voltage measured in one of the stations (A).

After the above conditions have been fulfilled, the distance-to-fault calculation is initiated.

In case of a fault 1, the following occurs:

Based on the current I_(A) measured in station A after the occurrence of the fault, R_(SC) and X_(SC) are determined, for example with FIG. 7, whereupon Z_(SC) in equation (2) may be determined.

the D₁ auxiliary vectors are calculated according to equation (10)

the coefficients a₁, b₁ and c₁ for the first distance-to-fault equation (9) are calculated according to equations (11), (12) and (13)

a first calculated value of the distance to the fault is calculated according to equation (14), and

a first value of the fault resistance is calculated according to equation (15).

The first calculated value of the distance to the fault is compared with the assumed value of the distance to the fault. If the difference between the values is greater than a pre-set maximally permissible difference value S₁, the assumed value in equations (2) and (7) is replaced by the first calculated value of the distance to the fault and the fault resistance.

Thereafter, a new calculation procedure is carried out according to the above, which provides new values of the distance to the fault and the fault resistance. If the difference between the last calculated value of the distance to the fault and the first calculated value is greater than the maximally permissible difference value, the first calculated value in equations (2) and (7) is replaced by the last calculated values of the distance to fault and the fault resistance.

This iteration continues until the difference value of two consecutive calculation procedures is smaller than the set maximally permissible difference value, whereupon the last calculated values of the distance to fault and the fault resistance are regarded as the real values.

In case of a fault 2, the following occurs:

Based on the current I_(A) measured in station A and an assumption of Z_(SC), after the occurrence of the fault, a preliminary value of the current through Z_(SC) is calculated by means of equation (5). By means of the preliminary value of the current, a first value of R_(SC) and X_(SC), that is also Z_(SC), may be obtained. The first calculated values are compared with the assumed value of Z_(SC). If the difference between the values is greater than a preset maximally permissible difference value S₂, the set value in equation (6) is replaced by the first calculated value of Z_(SC). This iteration continues until the difference value of two consecutive calculation procedures is smaller than the set maximally permissible difference value, whereupon the last calculated value of Z_(SC) constitutes the assumption of Z_(SC) which is needed in equations (3) and (16).

the D₂ auxiliary vectors are calculated according to equation (19)

the coefficients a₂, b₂ and c₂ of the second distance-to-fault equation (18) are calculated according to equations (20), (21) and (22)

a first calculated value of the distance to the fault is calculated according to equation (23), and

a first value of the fault resistance is calculated according to equation (22).

The first calculated value of the distance to the fault is compared with the set value of the distance to the fault. If the difference between the values is greater than a pre-set maximally permissible difference value S₃, the set value in equations (3) and (16) is replaced by the first calculated value of the distance to the fault and the fault resistance.

Thereafter, a new calculation procedure is carried out according to the above, which provides new values of the distance to the fault and the fault resistance. If the difference between the last calculated value of the distance to the fault and the first calculated value is greater than the maximally permissible difference value, the first calculated value in equations (3) and (16) is replaced by the last calculated values of the distance to fault and the fault resistance.

This iteration continues until the difference value of two consecutive calculation procedures is smaller than the set maximally permissible difference value, whereupon the last calculated values of the distance to fault and the fault resistance are regarded as the real values.

With access to the calculated values of the distances to fault, x₁₁ and x₂₁, and the corresponding fault resistances R_(f1) and R_(f2) and the equations described in the description, there is, according to the invention, a method for determining which of the distance-to-fault values represents the actual distance to the fault. The method is different depending on whether the fault is a three-phase fault or if it is an unsymmetrical fault, that is, if one or more phases is/are faultless.

If the fault is a three-phase fault, the following occurs:

Is R_(f1)>R_(f2)?Yes

If the answer is yes, it is assumed, as a first assumption, that the fault is a fault 2 and it is asked whether R_(f2)>R₀? where R₀ is a preset minimum value. If the answer is yes, the fault is a fault 2, that is, the calculated value of x₂₁ is the true value of the distance to the fault. If the answer is no, it is a fault 1, that is, the calculated value of x₁₁ is the true value of the distance to the fault.

Is R_(f1)>R_(f2)? No

If the answer is no, it is assumed, as a first assumption, that the fault is a fault 1 and it is asked whether R_(f1)>R₀? where R₀ is a preset minimum value. If the answer is yes, the fault is a fault 1, that is, the calculated value of x₁ is the true value of the distance to the fault. If the answer is no, it is a fault 2, that is, the calculated value of x₂₁ is the true value of the distance to the fault.

If the fault is an unsymmetrical fault, the currents through the fault resistances must first be determined for the two cases of fault.

For fault 1, the phase currents through the fault resistance may be determined with the aid of

i _(f(R,S,T)1) =i _(A(R,S,T)) +i _(B(R,S,T)1)  (25)

where

i_(A(R,S,T)) are the phase currents measured in station A

i_(B(R,S,T)1) is determined with the aid of equation (2) (i_(B)=I_(a-post))

and for fault 2, the phase currents through the fault resistance may be determined with the aid of

i _(f(R,S,T)2) ==i _(A(R,S,T)) +i _(B(R,S,T)2)  (26)

 where

 i_(B(R,S,T)2) is determined with the aid of equation (3) (i_(B=I) _(B-post))

The method for determining whether it is a question of a fault 1 or a fault 2 comprises, after a faultless phase has been identified in conventional manner, first determining the currents at the site of the fault in the faultless phase or phases for the two cases of fault according to equations (25), (2) and (26), (3), that is i_(ff1) och i_(ff2). Then follows the following procedure:

Is i_(ff1)>i_(ff2)?

Now, if i_(ff1)>i_(ff2) it is considered, as a first assumption, that the fault is a fault 2 and if i_(ff1)<i_(ff2) fit is considered, as a first assumption, that the fault is a fault 1.

The continued determination procedure is exactly identical with the procedure when the fault is a three-phase fault, that is: In case of a fault 2, it is asked whether R_(f2)>R₀? where R₀ is a preset minimum value. If the answer is yes, the fault is a fault 2, that is, the calculated value of x₂₁ is the true value of the distance to the fault. If the answer is no, it is a fault 1, that is, the calculated value of x₁₁ is the true value of the distance to the fault.

In case of a fault 1, it is asked whether R_(f1)>R₀? where R₀ is a preset minimum value. If the answer is yes, the fault is a fault 1, that is, the calculated value of x₁₁ is the true value of the distance to the fault. If the answer is no, it is a fault 2, that is, the calculated value of x₂₁ is the true value of the distance to the fault.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

A device for determining the distance from a station, at one end of a transmission line, until the occurrence of a fault on the transmission line according to the described method comprises certain measuring devices, measurement value converters, members for iterative treatment of the calculating algorithms of the method, a possibility of inputting to the member known conditions regarding the parameters of the power networks and assumed values of distance to fault, fault resistance, etc., a VDU indicating the calculated distance to fault, and a printer for printout of the calculated distance to fault. FIG. 8 largely shows an embodiment of the device. Embodiments similar to those of FIG. 8 are within the scope of the invention.

FIG. 8 shows a transmission line 1 between two stations A and B. The line is provided with capacitors, protecting against overvoltages, forming the impedance Z_(SC). Station A accommodates measuring devices 2 and 3 for continuous measurement of all the phase currents and phase voltages. In the measurement converters 4 and 5, a number of these consecutively measured values, which in case of a fault are passed to a calculating unit 6, are filtered and stored. The calculating unit is provided with the calculating algorithms described, programmed for the iterative processes which are needed for calculating the distance to fault and the fault resistance.

Even before a fault occurred on the transmission line, the calculating unit was provided with a number of conditions and assumptions to be able to perform the calculation of the distance to fault etc. These conditions comprise values of the emf and the impedance of both a forward and a backward power network. The calculating unit is also supplied with known values of the length and impedance of the transmission line, the distance from station A to the series capacitors, set values of the distance to fault, the fault resistance, Z_(SC) and the maximally permissible difference values S₁, S₂ and S₃. In connection with the occurrence of a fault, information about the type of fault may be supplied to the calculating unit for determining the fault-type matrix K_(f). In addition, after the occurrence of the fault, information is supplied as to whether the fault is positioned between the series capacitors and station B or whether the fault is positioned between station A and the series capacitors, that is, whether the fault is a fault 1 or a fault 2.

Then, when a fault has occurred, and the external conditions are fulfilled, the calculating procedure starts with the necessary iterations to obtain difference values which fall below the maximally permissible ones. The finally calculated values will then be transferred to a VDU which shows the distance to the fault and the magnitude of the fault resistance, respectively, for further actions. The device also comprises possibilities for printout of the result.

In the similar embodiments indicated above, parts of the embodiment according to FIG. 8 may be more or less integrated in the calculating unit. 

What is claimed is:
 1. A method for location of a fault, and determination of a resultant fault resistance, occurring on a power transmission line which is series-compensated with capacitors, with a parallel-connected overvoltage protection device, between two stations (A) and (B), wherein distance to the fault from station (A) and the fault resistance are determined by means of values measured in station (A) of a voltage before (pre) and a voltage and a current after (post) the occurrence of the fault, wherein, if the fault occurred between the series capacitors and station B, the distance to the fault is determined by the steps of: solving the expression: $x_{11} = \frac{{{Im}\left( {\underset{\_}{b}}_{1} \right)} - \sqrt{{{{Im}\left( {\underset{\_}{b}}_{1} \right)}{{Im}\left( {\underset{\_}{b}}_{1} \right)}} - {4{{Im}\left( {\underset{\_}{a}}_{1} \right)}{{Im}\left( {\underset{\_}{c}}_{1} \right)}}}}{2{{Im}\left( {\underset{\_}{a}}_{1} \right)}}$

which represents a first distance-to-fault equation; solving the expression: a ₁ x ₁ ² −b ₁ x ₁ +c ₁ −R _(f1)=0 for a, b and c, where a₁, b₁ and c₁ designate complex coefficients and x₁ represents the distance to the fault, and employing a first auxiliary vector D ₁=(Z _(A) +Z _(B) +Z _(L))(I _(A) _(—) _(post) I _(A) _(—) _(pre))+Z _(SC) _(—) _(post) I _(A) _(—) _(post) −Z _(SC) _(—) _(pre) I _(A) _(—) _(pre) Or where Z_(A) is a source impedance of a backward power network; Z_(B) is a source impedance of a forward power network; Z_(L) is an impedance of a power transmission line; Index, Post identifies a value after the occurrence of a fault; Index, Pre identifies a value before the occurrence of the fault; I_(A) is a fault current; Z_(S) is a complex impedance; and D₁ ^(T), which designates a transpose matrix of D₁, calculating a, b and c as follows: $\begin{matrix} {{\underset{\_}{a}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}Z_{L}K_{f}Z_{L}I_{A\_ post}}} & \quad \\ {{\underset{\_}{b}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}\quad \left( {{Z_{L}{K_{f}\left\lbrack {V_{A\_ post} + {\left( {Z_{L} - Z_{SC\_ post}} \right)I_{A\_ post}}} \right\rbrack}} + {Z_{B}K_{f}Z_{L}I_{A\_ post}}} \right)}} & (12) \\ {{\underset{\_}{c}}_{1} = {\frac{D_{1}^{T}}{D_{1}^{T}D_{1}}\left( {Z_{L} + Z_{B}} \right)\left( {V_{A\_ post} - {Z_{SC\_ post}I_{A\_ post}}} \right)K_{f}}} & (13) \end{matrix}$

and where Im(a₁), Im(b₁) and Im(c₁) are an imaginary part of the complex coefficients a, b₁ and c, respectively, and the fault resistance equals R _(f1) =Re( a ₁)x ₁₁ ² −Re( b ₁)x ₁₁ +Re( c ₁) where Re(a₁), Re(b₁) and Re(c₁) are a real part of the coefficients a ₁, b ₁ and c ₁.
 2. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 1, wherein: E_(A(R,S,R)) is the emf of the backward power network Z_(A(R,S,T)) is the source impedances of the backward power network; E_(B(R,S,T)) is the emf of the forward power network; Z_(B(R,S,T)) is the source impedances of the forward power network; Z_(L) is the impedance of the power transmission line; d is the distance in per unit from station A up to the series capacitor; dZ_(L) is the impedance of the power transmission line between station A and the series-compensated capacitor; and (1−d)dZ_(L) is the impedance of the power transmission line between the series capacitor and station B.
 3. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 1, wherein, when a fault has occurred, in order to characterize which type of fault has occurred, determining a fault-type matrix $K_{f} = \left\lbrack \begin{matrix} k_{RR} & k_{SR} & k_{TR} \\ k_{RS} & k_{SS} & k_{TS} \\ k_{RT} & k_{ST} & k_{TT} \end{matrix}\quad \right\rbrack$

according to the steps of: determinism elements k_(RR) . . . k_(TT) in K_(f) depending on the type of fault according to a procedure wherein, non-diagonal elements are given the value 0 if the phase in question is not concerned with the relevant fault and the value −1 if the phase is concerned with the relevant fault; and The diagonal elements are given the value 1 if the phase in question has a fault to ground at the fault in question and adding the sum of the absolute values of the non-diagonal elements in the relevant row.
 4. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 1, wherein a parallel connection of the series capacitors and the overvoltage protection device comprises a non-linear impedance Z_(SC) represented by a model equivalent impedance in the form of a series connection of a resistance R_(SC) and a reactance X_(SC), individually determining the reactance and resistance as a function of a traversing current, determining an actual value of resistance and reactance, employing actual current which, after the occurrence of the fault, flows through the impedance, such that a non-lineat impedance is set as Z _(SC) =R _(SC) +jX _(SC).
 5. A method for location of a fault, and determination of a resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 4, wherein, if the fault occurred between the series capacitors and station B, measuring the actual current in station A for determining the non-linear impedance.
 6. A method for location of a fault, and determination of a resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 5, wherein, if the fault occurred between station A and the series capacitors, an actual current I_(B) is used for determining the non-linear impedance said current I_(B) calculated in station B according to: I _(B)=((1−x ₂)Z _(L) +Z _(B) +Z _(SC))⁻¹.((Z _(A) +x ₂ Z _(L))I _(A)−(E _(A) −E _(B))) a value for assuming the non-linear impedance Z_(SC) and the distance to fault x₂, which current is used for determining a new value of Z_(SC) which is compared with the assumed value of Z_(SC), and if the difference between the assumed value of Z_(SC) and the new value is greater than a maximally permissible difference value S₂, the assumed value in I_(B) is replaced by the first calculated value of Z_(SC), repeating the previous calculation until the difference value of two consecutive calculating procedures is smaller than a maximum permissible difference value, whereupon a last obtained value of Z_(SC) constitutes an assumed value which is employed for a final determination of the distance to fault and the fault resistance.
 7. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 1, wherein an assumption is made on a distance to fault “x” from station A and a fault resistance R_(f).
 8. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 1, wherein, if the fault occurred between the series capacitors and station A, calculating an actual distance to fault and the fault resistance based on a calculating algorithm which comprises the equations E _(A) −E _(B) =[Z _(A) +Z _(B) +Z _(L) +Z _(SC) _(—) _(pre) ]I _(A) _(—) _(pre)  (1) E _(A) −E _(B) =[Z _(A) +x ₁ Z _(L) +Z _(SC) _(—) _(post) ]I _(A) _(—) _(post)−[(1−x ₁)Z _(L) +Z _(B) ]I _(B) _(—) _(post)  (2) V _(A) _(—) _(post) −V _(f)=(x ₁ Z _(L) +Z _(SC) _(—) _(post))I _(A) _(—) _(post) I _(f) =I _(A) _(—) _(post) +I _(B) _(—) _(post) where quantities with an index “pre” relate to the value prevailing before a fault has occurred and where quantities with an index “post” relate to the value prevailing immediately after the fault has occurred, and where equation (1) is a vectorial equation of a power transmission line corresponding to the conditions immediately before the fault has occurred, and where equation (2) is a vectorial equation of the power transmission line corresponding to the conditions immediately after the fault has occurred, and where V_(A) represents voltage measured in station A, and where V_(f) relates to the voltage on the power transmission line where the fault has occurred and which is determined with the aid of I _(f) =K _(f) V _(f)/R_(f) where I_(f) is the fault current through the fault resistance, and where x₁ is an unknown distance to fault from the power transmission line to the fault.
 9. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line according to claim 8, wherein, if the fault occurred between station A and the series capacitors, the calculation of the actual distance to fault and the fault resistance is based on a calculating algorithm which comprises the equations E _(A) −E _(B) [Z _(A) +Z _(B) +Z _(L) +Z _(SC) _(—) _(pre) ]I _(A) _(—) _(pre);  (1) E _(A) −E _(B) [Z _(A) +x ₂ Z _(L) ]I _(A) _(—) _(post)−[(1−x2)Z _(L) +Z _(B) +Z _(SC) _(—) _(post) ]I _(B) _(—) _(post);  (3) V _(A) _(—) _(post) −V _(f) =x ₂ Z _(L) I _(A) _(—) _(post); and I _(f) =I _(A) _(—) _(post) +I _(B) _(—) _(post); where quantities with an index “pre” relate to the value prevailing before a fault has occurred and where quantities with an index “post” relate to the value prevailing immediately after the fault has occurred, and where equation (1) is a vectorial equation of a power transmission line corresponding to a conditions immediately before the fault has occurred, and where equation (2) is a vectorial equation of the power transmission line corresponding to the conditions immediately after the fault has occurred, and where V_(A) represents a voltage measured in station A, and where V_(f) relates to a voltage on the power transmission line where the fault has occurred and which is determined with the aid of I _(f) =K _(f) V _(f) /R _(f)  (4) where I_(f) relates to the fault current through the fault resistance, and where x₂ designates an unknown distance to fault from the end point A of the power transmission line to the site of the fault.
 10. A device for carrying out the method for location of a fault, and determination of the resultant fault resistance, which has occurred on a series-compensated power transmission line between two stations A and B according to claim 1, wherein the device comprises measuring devices for continuous measurement of phase currents and phase voltages in one of the stations (A), measurement converters for filtering and storage of a number of these consecutively measured values, a calculating unit provided with the calculating algorithms of the method, including iteration possibilities, arranged for receiving signals in the form of measured, filtered and stored pre-fault phase currents and phase voltages as well as measured and filtered values of post-fault phase currents and phase voltages as well as signals indicating the emf (E_(A), E_(B)) and impedance (Z_(A), Z_(B)) of the backward and forward power network, the length (L) and impedance (Z_(L)) of the power transmission line, and the distance (d) from station A to the series capacitors, assumed values of the distance to fault (x) and the fault resistance (R_(f)), data for deterring the type of fault (K_(f)) and, pre-determined maximally permissible difference values (S₁, S₂, S₃).
 11. A method for location of a fault, and determination of the resultant fault resistance, which has occurred on a power transmission line which is series-compensated with capacitors, with a parallel-connected overvoltage protection device, between two stations (A) and (B), wherein the distance to the fault from one station (A) and the fault resistance are determined by means of values measured in said station (A) of voltage before (pre) and voltage and current after (post) the occurrence of the fault, wherein, if the fault occurred between station A and the series capacitors, the distance to the fault is determined by the steps of: solving the expression: $x_{21} = \frac{{{Im}\left( {\underset{\_}{b}}_{2} \right)} - \sqrt{{{Im}\left( {\underset{\_}{b}}_{2} \right){{Im}\left( {\underset{\_}{b}}_{2} \right)}} - {4{{Im}\left( {\underset{\_}{a}}_{2} \right)}{{Im}\left( {\underset{\_}{c}}_{2} \right)}}}}{2{{Im}\left( {\underset{\_}{a}}_{2} \right)}}$

which represents a second distance-to-fault equation  a ₂ x ₂ ² −b ₂ x ₂ +c ₂ −R _(f2)=0 for a₂, b₂ and C₂, where a₂, b₂ and c₂ designate complex coefficients and x₂ represents the distance to the fault; and employing a second auxiliary vector D ₂=(Z _(A) +Z _(B) +Z _(L))(I _(A) _(—) _(post) −I _(A) _(—) _(pre))+Z _(SC) _(—) _(post) I _(A) _(—) _(post) −Z _(SC) _(—) _(pre) I _(A) _(—) _(pre) Or where Z_(A) is a source impedance of a backward power network; Z_(B) is a source impedance of a forward power network; Z_(L) is an impedance of a power transmission line; Index, Post identifies a value after the occurrence of a fault; Index, Pre identifies a value before the occurrence of the fault; I_(A) is a fault current; Z_(S) is a complex impedance; and D₂ ^(T), which designates the transpose matrix of D₂, calculating the coefficients a₂, b₂ and c₂ as follows: $\begin{matrix} {{\underset{\_}{a}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}Z_{L}K_{f}Z_{L}I_{A\_ post}}} \\ {{\underset{\_}{b}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}\quad \left( {{Z_{L}K_{f}V_{A\_ post}} + {\left\lbrack {Z_{L} + Z_{SC\_ post} + Z_{B}} \right\rbrack K_{f}Z_{L}I_{A\_ post}}} \right)}} \\ {{\underset{\_}{c}}_{2} = {\frac{D_{2}^{T}}{D_{2}^{T}D_{2}}\left( {Z_{L} + Z_{B} + Z_{SC\_ post}} \right)K_{f}V_{A\_ post}}} \end{matrix}$

and where Im(a₂), Im(b₂) and Im(c₂) are an imaginary part of the complex coefficients a ₂, b ₂ and c ₂, and the fault resistance equals R _(f2) =Re( a ₂)x ₂₁ ² −Re( b ₂)x ₂₁ +Re( c ₂) where Re(a₂), Re(b₂) and Re(c₂) are a real part of the complex coefficients a₂, b₂ and c₂.
 12. A method for determining which of a located fault according to claim 2 represents the true location, wherein, if the located fault is a three-phase fault, determining whether R_(f1) is greater than R_(f2), and if no, it is assumed, as a first assumption, that the fault is an R_(f2) fault, and determining whether R_(f2)>R₀ where R₀ is a preset minimum value, and if so, the distance to fault is x₂₁, and if not, the distance to fault is x₁₁ and that if R_(f1) is smaller than R_(f2), it is assumed, as a first assumption, that the fault is an R_(f1) fault, and determining whether R_(f1)>R₀ and if so, the distance to fault is x₁, and if not, the distance to fault is x₂ if the located fault is an unsymmetrical fault, the phase currents through the fault resistance are first determined for a fault according to claim 1 according to i _(f(R,S,T)1) =i _(A(R,S,T)) +i _(B(R,S,T)1)  (25) where i_(A(R,S,T)) are the phase currents measured in station A i_(B(R,S,T)1) is determined by means of equation (2) (i_(B)=I_(B-post)) and for a fault according to i _(f(R,S,T)2) ==i _(A(R,S,T)) +i _(B(R,S,T)2) where i_(B(R,S,T)2) is determined by means of equation (3) (i_(B)=I_(B-post)) whereupon determining whether i_(f1) is greater than i_(f2), and if so, it is assumed, as a first assumption, that the fault is an R_(f2) fault, and determining whether R_(f2)>R₀ where R₀ is a preset minimum value, and if so, the distance to fault is x₂₁, and if not, the distance to fault is x₁₁ and if i_(f1) is smaller than i_(f2), it is assumed, as a first assumption, that the fault is an R_(f1) fault, and determining whether R_(f1)>R₀, and if so, the distance to fault is x₁, and if not, the distance to fault is x₂. 